506 Dr. L. Natanson on the Kinetic Energy 



Dissipation Function F defined as follows : — 



F= ip-pV)^ + (p-pv 2 )^ + {p-pP) ^7 



-*(g + Si-**.® * £)- *® + £)■ <«» 



we shall have 



*=- F+ KS + l + S> • • • < 31 > 



Returning to (29), multiplying by dx dy dz and integrating 



throughout the volume occupied by the medium, 



jr: f \T A dx dy dz = — Jj A {lu + m» + nw) d& 



+ 1 Jiff { p* §; (F + ^ 2 + F) + r, |, (F + ? + ? ) 



+ pr * % ^ + ^ + ^ } dx Ay As - (32) 



Here the direction-cosines of the normal to the element d$ of 

 the surface are denoted by Z, m, n. 



4. In order to interpret equation (32) let us adopt a some- 

 what generalized definition of " Kinetic Energy." Suppose 

 C (a vector) to represent any current or flux, and let q be its 

 velocity. We define then the kinetic energy per unit volume 

 to be the scalar product 



iS(Cq) = i(CV + C , V , + C'V // ) ? • - ■ "(33) 



Cj C", and C w being the components of C, and q 1 , q n , and q"' 

 the components of q. Thus, if C means an ordinary flux of 

 matter, of density p, then C=pq, and the "kinetic energy" 

 of matter, i. e. the kinetic energy in the case of a matter-flux, 

 is found as usually given. But we are now enabled to form 

 an idea of "kinetic energy" in other cases as well. Thus in 

 the case we are dealing with, the motion of a molecule 

 through space may be said to be equivalent to a " molecular 

 current " of the quantity Q carried about by the molecule ; 

 and then fQ, ??Q, and £Q will be the values of the compo- 

 - nents of that current. From (33) we conclude that the 

 kinetic energy of such a molecular current of Q is 



KF+V + m • "• • •'•'"■'". (34) 



